As shown in [1]}, with velocity birefringence, the different circular polarization modes will have different velocity phase velocities
\(v_A &\simeq & 1- \frac{1}{2} \rho _A \alpha _\mu (\tau ) \left(\frac{k}{aM_{\rm PV}}\right) \nonumber \\&=& 1 - \frac{1}{2} \rho _A ({\cal W}_1 - {\cal G}_1) \frac{k}{a}.\) 
